MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvreucsf Structured version   Unicode version

Theorem cbvreucsf 3318
Description: A more general version of cbvreuv 2947 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)
Hypotheses
Ref Expression
cbvralcsf.1  |-  F/_ y A
cbvralcsf.2  |-  F/_ x B
cbvralcsf.3  |-  F/ y
ph
cbvralcsf.4  |-  F/ x ps
cbvralcsf.5  |-  ( x  =  y  ->  A  =  B )
cbvralcsf.6  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvreucsf  |-  ( E! x  e.  A  ph  <->  E! y  e.  B  ps )

Proof of Theorem cbvreucsf
Dummy variables  v 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1678 . . . 4  |-  F/ z ( x  e.  A  /\  ph )
2 nfcsb1v 3301 . . . . . 6  |-  F/_ x [_ z  /  x ]_ A
32nfcri 2571 . . . . 5  |-  F/ x  z  e.  [_ z  /  x ]_ A
4 nfs1v 2147 . . . . 5  |-  F/ x [ z  /  x ] ph
53, 4nfan 1865 . . . 4  |-  F/ x
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
6 id 22 . . . . . 6  |-  ( x  =  z  ->  x  =  z )
7 csbeq1a 3294 . . . . . 6  |-  ( x  =  z  ->  A  =  [_ z  /  x ]_ A )
86, 7eleq12d 2509 . . . . 5  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  [_ z  /  x ]_ A ) )
9 sbequ12 1941 . . . . 5  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
108, 9anbi12d 705 . . . 4  |-  ( x  =  z  ->  (
( x  e.  A  /\  ph )  <->  ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph ) ) )
111, 5, 10cbveu 2299 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! z ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) )
12 nfcv 2577 . . . . . . 7  |-  F/_ y
z
13 cbvralcsf.1 . . . . . . 7  |-  F/_ y A
1412, 13nfcsb 3303 . . . . . 6  |-  F/_ y [_ z  /  x ]_ A
1514nfcri 2571 . . . . 5  |-  F/ y  z  e.  [_ z  /  x ]_ A
16 cbvralcsf.3 . . . . . 6  |-  F/ y
ph
1716nfsb 2151 . . . . 5  |-  F/ y [ z  /  x ] ph
1815, 17nfan 1865 . . . 4  |-  F/ y ( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )
19 nfv 1678 . . . 4  |-  F/ z ( y  e.  B  /\  ps )
20 id 22 . . . . . 6  |-  ( z  =  y  ->  z  =  y )
21 csbeq1 3288 . . . . . . 7  |-  ( z  =  y  ->  [_ z  /  x ]_ A  = 
[_ y  /  x ]_ A )
22 sbsbc 3187 . . . . . . . . 9  |-  ( [ y  /  x ]
v  e.  A  <->  [. y  /  x ]. v  e.  A
)
2322abbii 2553 . . . . . . . 8  |-  { v  |  [ y  /  x ] v  e.  A }  =  { v  |  [. y  /  x ]. v  e.  A }
24 cbvralcsf.2 . . . . . . . . . . . 12  |-  F/_ x B
2524nfcri 2571 . . . . . . . . . . 11  |-  F/ x  v  e.  B
26 cbvralcsf.5 . . . . . . . . . . . 12  |-  ( x  =  y  ->  A  =  B )
2726eleq2d 2508 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
v  e.  A  <->  v  e.  B ) )
2825, 27sbie 2105 . . . . . . . . . 10  |-  ( [ y  /  x ]
v  e.  A  <->  v  e.  B )
2928bicomi 202 . . . . . . . . 9  |-  ( v  e.  B  <->  [ y  /  x ] v  e.  A )
3029abbi2i 2552 . . . . . . . 8  |-  B  =  { v  |  [
y  /  x ]
v  e.  A }
31 df-csb 3286 . . . . . . . 8  |-  [_ y  /  x ]_ A  =  { v  |  [. y  /  x ]. v  e.  A }
3223, 30, 313eqtr4ri 2472 . . . . . . 7  |-  [_ y  /  x ]_ A  =  B
3321, 32syl6eq 2489 . . . . . 6  |-  ( z  =  y  ->  [_ z  /  x ]_ A  =  B )
3420, 33eleq12d 2509 . . . . 5  |-  ( z  =  y  ->  (
z  e.  [_ z  /  x ]_ A  <->  y  e.  B ) )
35 sbequ 2072 . . . . . 6  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
36 cbvralcsf.4 . . . . . . 7  |-  F/ x ps
37 cbvralcsf.6 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3836, 37sbie 2105 . . . . . 6  |-  ( [ y  /  x ] ph 
<->  ps )
3935, 38syl6bb 261 . . . . 5  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  ps ) )
4034, 39anbi12d 705 . . . 4  |-  ( z  =  y  ->  (
( z  e.  [_ z  /  x ]_ A  /\  [ z  /  x ] ph )  <->  ( y  e.  B  /\  ps )
) )
4118, 19, 40cbveu 2299 . . 3  |-  ( E! z ( z  e. 
[_ z  /  x ]_ A  /\  [ z  /  x ] ph ) 
<->  E! y ( y  e.  B  /\  ps ) )
4211, 41bitri 249 . 2  |-  ( E! x ( x  e.  A  /\  ph )  <->  E! y ( y  e.  B  /\  ps )
)
43 df-reu 2720 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
44 df-reu 2720 . 2  |-  ( E! y  e.  B  ps  <->  E! y ( y  e.  B  /\  ps )
)
4542, 43, 443bitr4i 277 1  |-  ( E! x  e.  A  ph  <->  E! y  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364   F/wnf 1594   [wsb 1705    e. wcel 1761   E!weu 2257   {cab 2427   F/_wnfc 2564   E!wreu 2715   [.wsbc 3183   [_csb 3285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-reu 2720  df-sbc 3184  df-csb 3286
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator